Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection

Abstract

<p style='text-indent:20px;'>This paper studies an <inline-formula><tex-math id="M1">\begin{document}$ (n+2) $\end{document}</tex-math></inline-formula>-dimensional nonlinear HIV dynamics model that characterizes the interactions of HIV particles, susceptible CD4<inline-formula><tex-math id="M2">\begin{document}$ ^{+} $\end{document}</tex-math></inline-formula> T cells and <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula>-stages of infected CD4<inline-formula><tex-math id="M4">\begin{document}$ ^{+} $\end{document}</tex-math></inline-formula> T cells. Both virus-to-cell and cell-to-cell infection modes have been incorporated into the model. The incidence rates of viral and cellular infection as well as the production and death rates of all compartments are modeled by general nonlinear functions. We have revealed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. The basic reproduction number <inline-formula><tex-math id="M5">\begin{document}$ \Re_{0} $\end{document}</tex-math></inline-formula> is determined which insures the existence of the two equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the model's equilibria. The global asymptotic stability of the two equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. We have proven that if <inline-formula><tex-math id="M6">\begin{document}$ \Re_{0}\leq1 $\end{document}</tex-math></inline-formula>, then the infection-free equilibrium is globally asymptotically stable, and if <inline-formula><tex-math id="M7">\begin{document}$ \Re _{0}>1 $\end{document}</tex-math></inline-formula>, then the chronic-infection equilibrium is globally asymptotically stable. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.</p>